To find the equations of the asymptotes for the given rational function:

$f(x) = \frac{x - 2}{4x - 5}$

Step 1: Vertical Asymptote

The vertical asymptotes occur when the denominator equals zero (and the numerator does not also equal zero at the same point). Set the denominator $4x - 5 = 0$ and solve for $x$:

$4x - 5 = 0 \quad \Rightarrow \quad x = \frac{5}{4}$

So, the vertical asymptote is:

$x = \frac{5}{4}$

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Step 2: Horizontal Asymptote

For horizontal asymptotes, we compare the degrees of the numerator and denominator:

• The degree of the numerator ($x - 2$) is 1.
• The degree of the denominator ($4x - 5$) is 1.

When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficients are:

$\text{Numerator coefficient: } 1 \quad \text{Denominator coefficient: } 4$

Thus, the horizontal asymptote is:

$y = \frac{1}{4}$

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Final Answer

The equations of the asymptotes are:

1. Vertical asymptote: $x = \frac{5}{4}$
2. Horizontal asymptote: $y = \frac{1}{4}$

Would you like a detailed graph of this function or additional explanation?

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Related Questions:

1. How do you determine the oblique asymptote of a rational function if applicable?
2. Can a function have more than one vertical asymptote?
3. What happens to $f(x)$ as $x \to \pm \infty$ for this specific function?
4. How do horizontal asymptotes differ from slant (oblique) asymptotes?
5. Can the graph of $f(x)$ intersect its horizontal asymptote?

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Tip:

When identifying asymptotes, always check for cancellations in the rational function to ensure there are no "holes" that might affect the vertical asymptote.